System for spectrum analysis including digital filter with variable sampling rate

ABSTRACT

A system for analyzing the frequency spectrum of an input signal includes a digital filter, the center frequency of which is varied by changing the sampling rate. The output signal of the digital filter is squared in a detector circuit; and the output signal of the detector circuit is fed to an integrator circuit. The integration time is varied inversely proportional to the center frequency of the digital filter to obtain a signal representative of the power spectrum of the input signal.

United States Patent 1191 Willett 1111 3,708,746 1451 Jan. 2, 1973 [54]SYSTEM FOR SPECTRUM ANALYSIS INCLUDING DIGITAL FILTER WITH VARIABLESAMPLING RATE 3,167,710 6/1965 COX 324/771 3,588,693 6/1971 Halleyetal.324/773 Primary Examiner-Stanley T. Krawczewicz [75] Inventor: RichardM. Willett, Boone, Iowa Au n j l [73] Assignee: Iowa State UniversityResearch 1 Foundation, Inc., Ames, Iowa [57] ABSTRACT [22] Filed: No 81971 A system for analyzing the frequency spectrum of an I input signalincludes a digital filter, the center PP N04 1961675 frequency of whichis varied by changing the sampling rate. The output signal of thedigital filter is squared in [52] U.S. Cl. ..324/77 D, 244/77 M, 324/78F, i a detector circuit; and the output signal of the detec- 323 1 7 torcircuit is fed to an integrator circuit. The integra- 51 1111.01. .001123/16 tion time is varied inversely p p a to the center 58 1 1611111156111611 ..324/77 E, 77 D, 77 B, 78 1), frequency of the digital filterto obtain a g l 324/78 E, 78 F, 78 J; 328/l67; 244/77 M representativeof the power spectrum of the input I 1 signal. 1 [56] References Cited 5Claims, 2 Drawing Figures UNITED STATES PATENTS 2,8Sl,6(1l 9/195811111111111 124/77 1:

II I 7 1/, 5 1 1 A W- A 7 BAND g'i'J t SQUARING INTEGRATING Fm 'F 'TQ'CONVERTER CIRCUIT CIRCUIT t TR R R R l3 CLOCK COUNTER RESET s PULSEGENERATOR CIRCUIT GENERATOR SYSTEM FOR SPECTRUM ANALYSIS INCLUDINGDIGITAL FILTER WITH VARIABLE SAMPLING RATE BACKGROUND AND SUMMARY Thepresent invention relates to a spectrum analyzer that is, a system fordetermining the frequency components of an input wave form. Typically,the input wave form is an analog electrical signal Spectrum analyzers ofthe type with which the present invention is concerned may be used indetermining the dynamic response of a flexible airframe, although it isnot so limited in its broader aspects. The dynamic response of theairframe of modern aircraft and missiles which operate over an extremelywide range of altitudes, velocity and load, may vary widely. Normally,the dynamic response of an airframe is given analytically in terms of atransfer characteristic or transfer function which, by a determinationof the poles and zeros thereof indicates the natural oscillatingfrequencies as well as the damping characteristics of the airframe. Thepoles of the transfer function due to bending of the airframe havedifferent natural frequencies, and they vary with time due to changes invelocity, altitude, on-board fuel, etc. Further, these poles are veryclose to the imaginary axis when plotted in the splane, indicating a lowdamping factor.

The airframes of aircraft and missiles of this type are stabilized bydetermining the exact oscillating mode of the airframe and then feedingsignals to the air-frame by means of a digital controller to stabilizeit. Such a system is disclosed in U. S. Pat. No. 3,572,618 of Richard M.Willett entitled Method for Stabilizing Aircraft and Missiles." Thisapplication discloses an adaptive control system for stabilizing anairframe wherein the actual airframe transfer function is continuouslydetermined during the flight; and this determination is used tostabilize the dynamics of the airframe as they actually exist. Ittherefore becomes necessary to determine the transfer function of theairframe in real time during flight; and this entails measuring thefrequency spectrum of signals picked up by sensors attached to theairframe and representative of the actual bending modes of the airframe.This application further discloses that the variation in sampling ratefor a digital filter may compensate for changes in the natural frequencyof the body bending modes of a flexible airframe. Hence, the polepositions of the bending poles can be kept at approximately the sameposition in the z-plane by varying the sampling rate of the digitalfilter, even though the naturalfrequency and position of these poles maychange considerably in the s-plane.

The present invention is directed to determining the process (i. e.,transfer function) of the bending-modes of a flexible airframe, and in abroader sense, in determining the frequency spectrum of an analogelectrical signal.

In the present invention, a narrow band digital filter receives theinput analog signal being analyzed and generates a signal representativeof the amplitude of the input signal within the narrow band of thefilter. The sampling rate of the filter is varied in a predeterminedmanner so that the bandwidth of the fllter'is swept through thefrequency spectrum of interest. Thus, as the sampling rate is changed,the output signal of the digital filter is representative of theamplitude of 2 the input signal for the various frequencies of interest.The output of the digital filter is then squared to generate signals,each representative of the power denwidth of the digital filter as itssampling frequency increases.

Persons skilled in the art will appreciated that the present inventionmay be employed-for uses other than the one disclosed from the followingdetailed description, accomplished by the attached drawing.

THE DRAWING FIG. 1 is a functional block drawing of a system constructedaccording to the present invention; and

FlG. 2 isa graph depicting the relative bandwidth of a digital filter asa function of the center frequency of the filter.

DETAILED DESCRIPTION Referring then to FIG. 1, reference numeral 10generally designates an input port or terminal on which there-is presenta signal, F (to), which is an analog electrical signal, the frequencyspectrum of which is desired to be analyzed. The signal is fed into anarrow band digital filter ll. 7 i

Digital filters are well-known in the art. They are real-timespecial-purpose digital computers which may be programmed so as toexhibit a desirable transfer characteristic. The co-efficients of thefilter for a desired function are simply entered into a programcontrolling the operation of the filter, and it is operated by means ofan external clock pulse generator 12 which 7 generates output pulsestransmitted along a line 13 to the digital filter 11 to control thesampling rate. That is, the sampling rate ofthe digital filter 11 isdetermined by the pulse rate of the clock pulse generator The pulse rateof the clock pulse generator 12, in turn, is controlled by a signal (aThe control signal m, is itself programmed to increase the pulse rate'ofthe block pulse generator 12 according to a predetermined pattern. lnthe case of analyzing the frequency spectrumof a flexible air frame, theinput signal F (w) may be generated by a transducer attached to theairframe and generating anelectrical signal proportional to the intensity of the instantaneous vibration of the airframe. The controlsignals to, may be generated by an 'onboard computer which also computesthe bending modes of the airframe.

A digital filter which may be used as the digital filter ll of thedisclosed embodiment is described in an article entitled ProgramableDigital Filter Performs'Multiple Functions by A. T. Anderson, inElectronics, Oct. 26, 1970.

The output of the digital filter 11, then, is a set of digital signalsrepresentative of the amplitude of the input wave form F (m) in thenarrow band of the filter; and it is fed to a digital to analogconverterlS for generating an analog signal representative thereof.Digital to analog converters are well-known in the art,

and any suitable such circuit may be employed in the present invention.The output signal of the digital to analog convertor 15 is then coupledto a squaring circuit 16 which may be an analog circuit which generatesan output signal representative of the square of the input signal. Atypical circuit used as the squaring circuit 16 may be a detectorcircuit used in conventional receivers for frequency modulated signals.Such circuits normally bias a semi-conductor diode in the region whereinthe current through it is proportional to the square of the voltageacross the diode terminals, thereby generating a current signalrepresentative of the square of the input voltage.

The output of the squaring circuit 16 is coupled to the input of ananalog integrating circuit 18. Analog integrating circuits are alsowell-known in the art, and they are used-commonly in analog computers.Alternatively, the squaring and integrating functions just mentionedcould be performed with digital circuits constructed oy techniques knownto persons skilled in this art.

The output signal of the clock pulse generator 12 is also coupled to adigital counter circuit 19 which may be a string of flip-flop circuitsconnected to form a serial counter to count the input pulses. The outputof the digital counter circuit 19 is connectedto the input of a resetgenerator circuit 20. When the digital counter circuit 19 reaches apredetermined count, representative of a predetermined number of outputpulses of the clock generator 12 (and, hence, related to a similarpredetermined number of sampling intervals for the digital filter 11),the reset generator 20 generates an output to reset the digital filter11, the digital to analog converter 15, the squaring circuit 16, and theanalog integrating circuit 18.

Digital filters, as in the case of a bandpass filter tuned to a discretefrequency, are completely analytically described by their transferfunctions. For a fixed location of poles and zeros in the z-plane, theproduct of the sampling period and the center frequency of the filter isa constant. That is to say,

w T=k 1) where m, is the center frequency ofthe digital filter,

T is the sampling period, and

k is a constant.

It will be noted from the above equation that if the sampling rate ofthe filter is changed, its center frequency will also be changed. Hence,the spectrum analyzer of the present invention, in operation, uses thisvariation of the sampling rate to sweep the center frequency of thedigital filter through the desired frequency spectrum of interest. Thisis done, of course, in discrete steps. That is to say, referring to FIG.2, if the sampling period is 7], the center frequency of the filter is mand the bandpass of the digital filter is diagrammatically illustratedby the graph 23. If the period of sampling is decreased (that is, thesampling frequency is increased) to a value T then the center frequencyof the digital filter 11 will be correspondingly increased to (0.according to the relationship of equation 1 above. The new bandpass ofthe digital filter is diagrammatically illustrated by the curve 24.

As mentioned, the output signal of the digital filter is representativeof amplitude, and after it is squared in the squaring circuit 16, it isrepresentative of the power spectrum of the signal within the bandpassof the filter. The output signal of the squaring circuit is then in- 5tegrated for a fixed number of samples to generate a signalrepresentative of the power spectral density of the input signal, F (m)at the center frequency of the filter. That is to say,

onef pow er) (2) where The Q (a figure of merit) of a bandpass filter isdefined as the ratio of its center frequency, w,., to the differencebetween frequencies at the half-power points on its bandwidth curve,denoted Aw for the curve 24 of HO. 2.

For each fixed location of poles and zeros, a filter has a fixed andconstant value of 0 whether it is a digital filter or a lumped constantparameter filter. As the center frequency of the filter changes, thebandwidth of the filter changes proportionally. That is to say, as thecenter frequency of the digital filter 11 changes from (u of FIG. 2 to mthe bandwidth of the curve 23 changes proportionally. Hence, the energycontent of the output signal of the digital filter will depend upon thenumber of frequency components of the input signal which are presentwithin the frequency range of the filter. Since the bandwidth of thefilter is higher when the center frequency is raised, the energy contentof the output signal will also be higher provided that the input signalhas a constant spectral density function, for example, white noise, asis the case in bending modes for an airframe.

The present invention compensates for the changing bandwidth of thefilter characteristic by changing the integration time of theintegrating circuit 18 inversely proportional to the center frequency ofthe digital filter 11. That is to say, the digital counter circuit 19actuates the reset generator 20 after a preset numberof clock pulses aresensed, irregardless of the repetition rate of those clock pulses.Hence, as the center frequency of the digital filter 11 is increased byincreasing the sampling rate (which is the pulse rate of the clock pulsegenerator 12) the integrating time is correspondingly reduced becausethe integration time is determined by the number of clock pulses, not apreset time.

The bandwidth ofa bandpass filter in spectrum analysis plays a veryimportant role. In order to obtain an accurate power spectral densityanalysis of the input signal by a digital filter, a filter having anarrow bandwidth is preferable to one having a relatively widebandwidth. However, the response time of the filter increases as itsbandwidth decreases. Thus, an engineering trade-off must be made indesigning the filter.

In one computer simulation of a typical problem, the bandwidth of thebending mode of an airframe was determined from a prior analysis, and asecond-order filter-was considered to be narrow enough to be suitable.The final transfer function of the desired bandpass digital filter wasderived from a conventional continuwill be able to modify certain of thestructure which has ous, second-order Butterworth low-pass filter.Transfer been illustrated and to substitute equivalent elements functionof this lowpass filter (normalized to one radian for those disclosedwhile continuing to practice the per second) was determined to be:principle of the invention; and it is, therefore, intended 5 that allsuch modifications and substitutions be covered 1 (s+0.7071068 07071068)(s+0.7071068 07071068) zggifig fg gs wthm the Spmt and Scope of thelclaim: Transformation from a continuous lowpass filter to a 10continuous bandpass filter was then made by using the 1. A system foranalyzing the spectrum of an input relation time-varying electricalsignal comprising: digital filter means receiving said input signal andgenerating an S'=(B/2) Pi'HwPiw/Z) (4) output signal representative ofthe amplitude of said where A input signal within a predeterminedbandwidth, said was the deslred bahdwldth of the cohhhuous band filtermeans being characterized in that the center P hhel" frequency of saidfilter varies with the sampling rate, "r was the desired centerfrequency of the cohhhu' said filter further being adapted to have avariable sam- 0115 bandpass filter, pling rate; squaring circuit meansreceiving the output Pi was the real P of the i Foot of the Continuoussignal of said digital filter means for generating a signal lowpassfilter, and representative of the square of said output signal and Q,-was the imaginary part of the i root of the conthereby representative ofthe power spectrum of said tinuous lowpass filter. input signal; meansfor varying the sample rate of said In our case, an arbitrary choice ofbandwidth and digital filter means; and counter circuit means for centerfrequency was made. A bandwidth of 0.20 radi- 5 decreasing theintegration time of said integration cirans/second was chosen at thecenter frequency of 12 cuitmeans as said sampling rate increases.radians/second of the desired Butterworth bandpass 2. The system ofclaim 1 wherein said counter circuit filter. The transfer function ofthe equivalent fourthmeans inCllldeS a digital COllflter C t ting theorder bandpass filter from the lowpass filter for the p g ram Pulses forresetting Said integration above-mentioned bandwidth and centerfrequency was cult means when a predetermined "umber of Sample f d to brate pulses have been counted.

1 (s+0.07071068:kj12.07071068)(S+0.0707l068 ijllli202 ll32 The transferfunction of the bandpass filter in z- 3. The system of claim 1 whereinsaid digital filter domain was computer by the computer program means ischaracterized in having a relatively narrow CNVRT disclosed in thethesis of R. M. Willett, Sambandpass.

ple-Data Control Utilizing Variable Sampling Rate," 4. A system fordetermining the frequency comil bl i h lib f I St t U iv i at ponentsofan input electrical signal comprising: narrow Ames, Iowa (1966), andis given in equation (6). The band digital filter means receiving saidinput signal for sampling time of 0.0436 seconds was chosen for thegenerating an Output Signal representative of the greater frequency, wof the filter at 12 radians/second. frequency p n n f Said input waveform within 0.03045239[2(z0.9220736) (z- 1.0s4s37n D (z) (2-0.8620185ij0.5007765) (2 0S650800ij04954518) The frequency response of the filterwas computed the narrow band of said digital filter means; squaring withthe help of a computer program given in Appendix circuit means receivingoutput signals of said digital C of the Master's Thesis of R. P. S.Bhatia, An Infilter means for generating a signal representative ofvestigation of Spectrum Analysis by Means of a Digital the squarethereof; integrator circuit means receiving Filter with VariableSampling Rate, available in the the output Signal of said squaringcircuit means for inlib f I S t U i it A tegrating the same ovcr apredetermined time; means (N b |97 I order to id h ff f for varying thesampling rate ofsaid digital filter means frequency aliasing thefrequency range for this compu- Q thereby F hgf the Center frequency Ofe handtation was chosen such, that only one peak occurred in ,0 Width ofSald hg filter t; and er clrcult that range. means responsive to thesampling rate of said digital A a genera| rule h Sample rate orfrequency filter means for resetting said integrating circuit meansshould always be at least twice the bandwidth of the after apredetermmed number of Sample ratchulflcs input signal being analyzedfor the lowest sample h hif as the center h q y 0f h bandwidth offrequency Mumple digital filters may be used Covering said digitalfilter means increases, the integration time diff frequency ranges if ii not of said integrating circuit means decreases.

Having thus described in detail a preferred embodi- 5. in a method ofdetermining the frequency specment of the inventive system, personsskilled in the art trum of an input signal, the steps comprisingprocessing decreasing the integration time of said squared signal togenerate a set of signals representative of the energy content of saidinput signal at predetermined bandwidths over a spectrum.

1. A system for analyzing the spectrum of an input time-varyingelectrical signal comprising: digital filter means receiving said inputsignal and generating an output signal representative of the amplitudeof said input signal within a predetermined bandwidth, said filter meansbeing characterized in that the center frequency of said filter varieswith the sampling rate, said filter further being adapted to have avariable sampling rate; squaring circuit means receiving the outputsignal of said digital filter means for generating a signalrepresentative of the square of said output signal and therebyrepresentative of the power spectrum of said input signal; means forvarying the sample rate of said digital filter means; and countercircuit means for decreasing the integration time of said integrationcircuit means as said sampling rate increases.
 2. The system of claim 1wherein said counter circuit means includes a digital counter circuitcounting the sampling rate pulses for resetting said integration circuitmeans when a predetermined number of sample rate pulses have beencounted.
 3. The system of claim 1 wherein said digital filter means ischaracterized in having a relatively narrow bandpass.
 4. A system fordetermining the frequency components of an input electrical signalcomprising: narrow band digital filter means receiving said input signalfor generating an output signal representative of the frequencycomponents of said input wave form within the narrow band of saiddigital filter means; squaring circuit means receiving output signals ofsaid digital filter means for generating a signal representative of thesquare thereof; integrator circuit means receiving the output signal ofsaid squaring circuit means for integrating the same over apredetermined time; means for varying the sampling rate of said digitalfilter means to thereby change the center frequency of the bandwidth ofsaid digital filter means; and counter circuit means responsive to thesampling rate of said digital filter means for resetting saidintegrating circuit means after a predetermined number of sample ratepulses whereby as the center frequency of the bandwidth of said digitalfilter means increases, the integration time of said integrating circuitmeans decreases.
 5. In a method of determining the frequency spectrum ofan input signal, the steps comprising processing said signal in a narrowband digital filter, squaring the output signal of said digital filter,then integrating said squared signal, and incrementally increasing thesampling rate of digital filter while correspondingly decreasing theintegration time of said squared signal to generate a set of signalsrepresentative of the energy coNtent of said input signal atpredetermined bandwidths over a spectrum.